Physical Review Letters

Solution and Critical Behavior of Some "Three-Dimensional" Ising Models with a Four-Spin Interaction

Masuo Suzuki

Published Feb 21, 1972• 73 citations

Abstract

An exact solution is obtained for some "three-dimensional" Ising models with a four-spin interaction. The spontaneous magnetization is vanishing for all temperatures. The temperature derivative of the susceptibility shows a divergence of the form ${({T}_{c}\ensuremath{-}T)}^{\frac{\ensuremath{-}7}{8}}$ below the critical point. The specific heat shows a logarithmic singularity. Several other related models are introduced with new types of singularities.

DOI: 10.1103/physrevlett.28.507

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High-temperature critical properties of the Ising model on a triple of related lattices

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Physical Review Letters

Solution and Critical Behavior of Some "Three-Dimensional" Ising Models with a Four-Spin Interaction

Masuo Suzuki

Published Feb 21, 1972• 73 citations

Publisher

Abstract

{({T}_{c}\ensuremath{-}T)}^{\frac{\ensuremath{-}7}{8}}

below the critical point. The specific heat shows a logarithmic singularity. Several other related models are introduced with new types of singularities.

DOI: 10.1103/physrevlett.28.507

Related papers

High-temperature critical properties of the Ising model on a triple of related lattices

Joseph Allen Leu, Donald D. Betts, C. J. Elliott

Three regular three-dimensional lattices of coordination numbers, q = 3, 4, and 6 are introduced. Exact relations are derived among the specific-heat singularity amplitudes and among the susceptibility singularity amplitudes. Exact high-temperature series expansions for the partition function and the susceptibility are derived for the q = 3 and q = 6 lattices. Precise values of the critical temperature, susceptibility amplitude, critical energy, and critical entropy are obtained for all three lattices. The variation of Ising critical parameters with coordination number is discussed.Three regular three-dimensional lattices of coordination numbers, q = 3, 4, and 6 are introduced. Exact relations are derived among the specific-heat singularity amplitudes and among the susceptibility singularity amplitudes. Exact high-temperature series expansions for the partition function and the susceptibility are derived for the q = 3 and q = 6 lattices. Precise values of the critical temperature, susceptibility amplitude, critical energy, and critical entropy are obtained for all three lattices. The variation of Ising critical parameters with coordination number is discussed.

Canadian Journal of PhysicsAug 15, 196921 citations

Critical Behavior of the Four-Dimensional Ising Ferromagnet and the Breakdown of Scaling

M A Moore

Physical review. B, Solid stateMar 1, 197035 citations

Critical exponents of four-dimensional random-field Ising systems

Alexander K. Hartmann

The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to ${32}^{4}$ are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness ${h}_{c}=4.18(1)$ and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to ${32}^{4}$ are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness ${h}_{c}=4.18(1)$ and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$

arXiv:cond-mat/0201441Physical review. B, Condensed matterMay 7, 200230 citations

Ising-Model Critical Indices below the Critical Temperature

George Allen Baker Jr., David S. Gaunt

We review the estimates for the critical indices of the Ising model below ${T}_{C}$ and conclude (for 3 dimensions): ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.066_{\ensuremath{-}0.04}^{}{}_{}{}^{+0.16}$, $\ensuremath{\beta}=0.312_{\ensuremath{-}0.005}^{}{}_{}{}^{+0.002}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.310_{\ensuremath{-}0.05}^{}{}_{}{}^{+0.04}$ for the specific heat, magnetization, and magnetic susceptibility, respectively. For 2 dimensions, we estimate ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75_{\ensuremath{-}0.00}^{}{}_{}{}^{+0.01}$. In order to explain previous estimates of ${\ensuremath{\alpha}}^{\ensuremath{'}}=0$, we point out that a low power can, in practice, look deceptively like a logarithm. Finally, we discuss the behavior of the specific heat at constant magnetization.We review the estimates for the critical indices of the Ising model below ${T}_{C}$ and conclude (for 3 dimensions): ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.066_{\ensuremath{-}0.04}^{}{}_{}{}^{+0.16}$, $\ensuremath{\beta}=0.312_{\ensuremath{-}0.005}^{}{}_{}{}^{+0.002}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.310_{\ensuremath{-}0.05}^{}{}_{}{}^{+0.04}$ for the specific heat, magnetization, and magnetic susceptibility, respectively. For 2 dimensions, we estimate ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75_{\ensuremath{-}0.00}^{}{}_{}{}^{+0.01}$. In order to explain previous estimates of ${\ensuremath{\alpha}}^{\ensuremath{'}}=0$, we point out that a low power can, in practice, look deceptively like a logarithm. Finally, we discuss the behavior of the specific heat at constant magnetization.

Physical ReviewMar 10, 196774 citations

Exact solution in an external magnetic field of Ising models with three-spin interactions

Luke Lokia Liu, Eugene Stanley

The three-spin interaction Hamiltonian ${\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}$ (the $s$ and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g. $x$ are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite $T$ , because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.The three-spin interaction Hamiltonian ${\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}$ (the $s$ and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.$x$ are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite $T$, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.

Physical review. B, Solid stateOct 1, 197417 citations

New singularities in the critical behavior of random Ising models at marginal dimensionalities

Amnon Aharony A

Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at $d=4$ , and (ii) dipolar interactions at $d=3$ . In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat $C$ are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$. $D$ is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of $C$ and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at $d=4$, and (ii) dipolar interactions at $d=3$. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat $C$ are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$. $D$ is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of $C$ and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.

Physical review. B, Solid stateMar 1, 1976111 citations

{({T}_{c}\ensuremath{-}T)}^{\frac{\ensuremath{-}7}{8}}

below the critical point. The specific heat shows a logarithmic singularity. Several other related models are introduced with new types of singularities.

The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to

{32}^{4}

are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness

{h}_{c}=4.18(1)

and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$

We review the estimates for the critical indices of the Ising model below

{T}_{C}

and conclude (for 3 dimensions): ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.066_{\ensuremath{-}0.04}^{}{}_{}{}^{+0.16}$, $\ensuremath{\beta}=0.312_{\ensuremath{-}0.005}^{}{}_{}{}^{+0.002}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.310_{\ensuremath{-}0.05}^{}{}_{}{}^{+0.04}$ for the specific heat, magnetization, and magnetic susceptibility, respectively. For 2 dimensions, we estimate ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75_{\ensuremath{-}0.00}^{}{}_{}{}^{+0.01}$. In order to explain previous estimates of ${\ensuremath{\alpha}}^{\ensuremath{'}}=0$, we point out that a low power can, in practice, look deceptively like a logarithm. Finally, we discuss the behavior of the specific heat at constant magnetization.

s

and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.

x

are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite

T

, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.

Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at

d=4

, and (ii) dipolar interactions at

d=3

. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat

C

are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$.

D

is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of

C

and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.